K-theory of endomorphisms via noncommutative motives

In this article we study the K-theory of endomorphisms using noncommutative motives. We start by extending the K-theory of endomorphisms functor from ordinary rings to (stable) infinity categories. We then prove that this extended functor KEnd(-) not only descends to the category of noncommutative motives but moreover becomes co-represented by the noncommutative motive associated to the tensor algebra S[t] of the sphere spectrum S. Using this co-representability result, we then classify all the natural transformations of KEnd(-) in terms of an integer plus a fraction between polynomials with constant term 1; this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative co-algebra structure of S[t], we explain how the (rational) Witt vectors can also be recovered from the symmetric monoidal category of noncommutative motives. Along the way we show that the K_0-theory of endomorphisms of a connective ring spectrum R equals the K_0-theory of endomorphisms of the underlying ordinary ring \pi_0(R).